Information Technology Reference
In-Depth Information
μ
σ
The mean
and variance
vectors are estimated for each iteration
t
considering
=
,...,
j
1
n
parameters (e.g. if the controller has two parameters, the input scaling
factors
K
e
,
K
ce
, then
n
=
2) as:
N
elite
x
ij
N
elite
μ
tj
=
(7.11)
i
=
1
x
ij
−
μ
tj
2
N
elite
N
elite
σ
tj
=
(7.12)
i
=
1
N
elite
where 4
20
A very useful criterion that penalizes long duration transients is known as
ITAE
;
the integral of time multiplied by the absolute value of error:
t
=
|
(
)
|·
ITAE
e
t
tdt
(7.13)
0
This performance index is considered in the optimization problem (
7.13
). The
ITAE
performance index is more selective than others such as the integral of absolute
error (
IAE
) and the integral of square error (
ISE
).Moreover, the ITAE criterion usually
produces smaller overshoots and oscillations than the
ISE
and the
IAE
.
Another performance index considered in this chapter is the integral of squared
time multiplied by square error (
ISTSE
). Sudden changes in load disturbances pro-
duce large errors. Therefore, it is useful to have a criterion that places little weight
on initial errors and that heavily penalises errors occurring late in time. The
ISTSE
criterion is also, in many cases, a natural choice, at least for making a non-oscillatory
and overshoot-free response.
t
e
2
t
2
dt
ISTSE
=
(
t
)
·
(7.14)
0
The
ITAE
criterion is applied in the CE algorithm as a cost function because
it provides good selectivity among the above-mentioned performance indices. The
CE algorithm starts with initial fuzzy controller parameters
K
FCS
K
e
K
ce
]
i
and
evaluates the
ITAE
performance index (
7.13
). For the sake of clarity, Eq. (
7.9
)is
particularised as follows:
=[
K
FCS
=[
K
e
,
K
ce
]
OPT
=
argmin
(
ITAE
)
(7.15)
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